Hi J.,
I've never cleaned a car seat like this, but a 1/2 vinegar and 1/2 water solution removes pet odors (not stains, though) from carpet. I'd try it on a hidden area to test for discoloration. It sounds like your seat cover is porous, explaining why it smells bad but looks clean? You could saturate the seat with the solution and then use your steam cleaner to remove the liquid.
p.s. Vinegar is my new favorite cleaning solution. Adding baking soda and hydrogen peroxide to your arsenal, any type dirt/lime/spills/stains have no chance! (I really like Zout, too, it's magic for removing laundry stains.)